## Sunday, June 24, 2012

### Chappell Master Framing Square Unequal Pitched Roof Review

In this blog/article I'm taking the Chappell Master Framing Square for a test drive for accuracy using the Unequal Pitched Roof Rafter Tables imprinted on the stainless steel carpenters framing square.

We'll use an Eave angle of 90° with an 12:12 major pitch (B)and an minor pitch of 9:12 (A).

```
```
`This drawing is a French Compagnon Roof Framing Kernel drawn in 2D and 3D. The 2D geometric drawing of the French Compagnon Roof Framing Kernel is two tetrahedrons unfolded. Checking the 2D & 3D geometric drawing angles against the Chappell Master Framing Square clearly shows how accurate the unequal pitched roof rafter tables are on the Chappell Master Framing Square. This unequal pitched roof is an 12/12 and 9/12 with an eave angle of 90°.`
```
```
` `
```
```

Major Pitch 12/12
Minor Pitch 9/12
Major Plan Angle = 36.86990° tan: 0.75
Minor Plan Angle = 53.13010° tan: 1.333333
Major Pitch Angle = 45.00000° 12/12
Minor Pitch Angle = 36.86990° 9/12
Hip Rafter Angle = 30.96376° tan: 0.6 (line 1 on the square)
Major Hip Rafter Backing Angle = 34.44990° (line 9 on the square)
Minor Hip Rafter Backing Angle = 21.10020° (line 9 on the square)
Major Hip Rafter Sidecut Angle = 48.82567° tan: 1.143324
Minor Hip Rafter Sidecut Angle = 32.74587° tan: 0.64312
Major Jack Rafter Sidecut Angle = 43.31386° tan: .942809 (line 4 on the square)
Minor Jack Rafter Sidecut Angle = 30.96376° tan: 0.6 (line 4 on the square)
Major Roof Sheathing Angle = 46.68614° tan: 1.06066 (line 5 on the square)
Minor Roof Sheathing Angle = 59.03624° tan: 1.666666 (line 5 on the square)
Major Hip Rafter Square Tail Saw Bevel Angle = 34.44990° (line 9 on the square)
Minor Hip Rafter Square Tail Saw Bevel Angle = 28.685° (line 9 on the square)
Major Hip Rafter Square Tail Miter Angle = 28.07248° tan: 0.5333 (line 8 on the square)
Minor Hip Rafter Square Tail Miter Angle = 12.18863° tan: 0.216 (line 8 on the square)
Major Purlin Miter Angle = 43.31385° tan: 0.942809 (line 7 on the square)
Minor Purlin Miter Angle = 24.22774° tan: 0.45 (line 7 on the square)

#### The only roof framing angle missing on the Chappell Master Framing Square is the hip rafter side cut angle. However, you can set your saw blade bevel angle to the plan angle of the roof pitch to produce the side cut angle on the hip rafter.

This drawing is a tetrahedron slice of the hip rafter side cut angle at the ridge for the 12/12 side of the hip rafter.
```
```

``` Major Pitch 12/12
Minor Pitch 9/12
Major Plan Angle = 36.86990° tan: 0.75
Major Pitch Angle = 45.00000° 12/12
Hip Rafter Angle = 30.96376° tan: 0.6
Major Hip Rafter Backing Angle = 34.44990°
Major Hip Rafter Sidecut Angle = 48.82567° tan: 1.143324```
```
```
```
```
```
```

This drawing is a tetrahedron slice of the hip rafter side cut angle at the ridge for the 9/12 side of the hip rafter.

Major Pitch 12/12
Minor Pitch 9/12
Minor Plan Angle = 53.13010° tan: 1.333333
Minor Pitch Angle = 36.86990° 9/12
Hip Rafter Angle = 30.96376° tan: 0.6
Minor Hip Rafter Backing Angle = 21.10020°
Minor Hip Rafter Sidecut Angle = 32.74587° tan: 0.64312

Note: All drawings in this post were generated using my Google Sketchup Roof Framing Geometry plugin.
```

Checking the trigonometric calculations below against the Chappell Master Framing Square clearly shows how accurate the unequal pitched roof rafter tables are on the Chappell Master Framing Square. Whether your a stick framer or a timber framer.
Tetrahedron Angles for 12/12 irregular hip roof

D Angle = 36.8699
A Angle = 45.0000
C Angle = 30.9638
E Angle = 43.3139
B Angle = 34.4499

90-D Angle = 53.1301
90-A Angle = 45.0000
90-C Angle = 59.0362
90-E Angle = 46.6861
90-B Angle = 55.5501

The dihedral angle is the angle measured between two planes.
Dihedral Angle = 111.1002

Polygonal Pyramid Sides = 3.39
Pyramid Slant Angle = 45
Pitch Angle = 45
Plan Angle  = 36.8699
Working Angle = 53.0973451327434

Hip Rafter Pitch Angle = arctan( tan( Pitch Angle ) * sin( Plan Angle ))
Hip Rafter Pitch Angle = arctan( tan( 45 ) * sin( 36.8699 )) = 30.9637579168711°

Hip Rafter Backing Angle = arctan( sin( Hip Rafter Pitch Angle) ÷ tan( Plan Angle ) )
Hip Rafter Backing Angle = arctan( sin( 30.9637579168711) ÷ tan( 36.8699 ) ) = 34.4499007767502°

Hip Rafter Plumb Backing Angle = arctan( tan( Pitch Angle ) * cos( Plan Angle ))
Hip Rafter Plumb Backing Angle = arctan( tan( 45 ) * cos( 36.8699 )) = 38.6598073928135°

Hip Rafter Side Cut Angle  = arctan( cos( Hip Rafter Pitch Angle ) ÷ tan( Plan Angle ))
Hip Rafter Side Cut Angle = arctan( cos( 30.9637579168711 ) ÷ tan( 36.8699 )) = 48.825665948811°

Jack Rafter Side Cut = arctan( cos( Pitch Angle ) ÷ tan( Plan Angle ))
Jack Rafter Side Cut = arctan( cos( 45 ) ÷ tan( 36.8699 )) = 43.3138542102836°

Roof Sheathing Angle = arccos( cos( Plan Angle ) * cos( Hip Rafter Pitch Angle ))
Roof Sheathing Angle = arccos( cos( 36.8699 ) * cos( 30.9637579168711 )) = 46.6861457897164°

Frieze Block Saw Miter Angle = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Frieze Block Saw Miter Angle = arctan( sin ( 45 ) ÷ tan( 36.8699 )) = 43.3138542102836°

Frieze Block Saw Blade Bevel Angle = arctan( sin( Frieze Block Saw Miter Angle ) ÷ tan( Pitch Angle ))
Frieze Block Saw Blade Bevel Angle = arctan( sin( 43.3138542102836 ) ÷ tan( 45 )) = 34.4499007767502°

Square Tail Fascia Miter Angle  = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Square Tail Fascia Miter Angle  = arctan( sin ( 45 ) ÷ tan( 36.8699 )) = 43.3138542102836°

Square Tail Fascia Saw Blade Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Square Tail Fascia Saw Blade Bevel Angle  = arcsin( cos ( 45 )  *  cos( 36.8699 )) = 34.4499007767502°

Hip Rafter Square Tail Miter Angle  = arctan (( cos( Pitch Angle) * sin( Pitch Angle ) * cos(Plan Angle)) ÷ tan( Plan Angle))
Hip Rafter Square Tail Miter Angle  = arctan (( cos( 45) * sin( 45 ) * cos( 36.8699 )) ÷ tan(36.8699)) = 28.0724841662578°

Hip Rafter Square Tail Saw Blade Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Hip Rafter Square Tail Saw Blade Bevel Angle  = arcsin( cos ( 45 )  *  cos( 36.8699 )) = 34.4499007767502°

90° - Hip Rafter Square Tail Side Cut Angle (Foot)
R3 equals P2 rotated by C5.
R3 equals Jack Rafter Side Cut Angle rotated by the Hip Rafter Backing Angle
R3 = arctan( cos ( Hip Rafter Backing Angle ) * tan( Jack Rafter Side Cut Angle ))
R3 = arctan( cos ( 34.4499007767502 ) * tan( 43.3138542102836 )) = 37.8636443877934°

Dormer Square Tail Fascia Miter Angle  = arctan( cos ( Dormer Pitch Angle ) ÷ tan( Main Roof Pitch Angle ))
Dormer Square Tail Fascia Miter Angle  = arctan( cos ( 45 ) ÷ tan( 45 )) = 35.2643896827547°
Dormer Square Tail Fascia Miter Angle on Fascia = 90° - 35.2643896827547° = 54.7356103172453°

Dormer Square Tail Saw Blade Bevel Angle  = arcsin( sin ( Dormer Pitch Angle ) *  cos( Main Roof Pitch Angle ))
Dormer Square Tail Saw Bevel Angle  = arcsin( sin ( 45 )  *  cos( 45 )) = 30°

Rake Stud Saw Miter Angle = Hip Rafter Plumb Backing Angle = 38.6598073928135°
Rake Stud Saw Blade Bevel Angle = arctan(cos(Hip Rafter Plumb Backing Angle)* tan(Hip Rafter Pitch Angle))
Rake Stud Saw Blade Bevel Angle = atan(cos(38.6598073928135)* tan(30.9637579168711)) = 25.1040917208509°

Rake Plate Miter Angle = Jack Rafter Side Cut Angle = 43.3138542102836°
Rake Plate Saw Blade Bevel Angle = Roof Pitch Angle = 45°

Purlin Miter Angle  = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Purlin Miter Angle  = arctan( sin ( 45 ) ÷ tan( 36.8699 ))= 43.3138542102836°

Purlin Saw Blade Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Purlin Saw Blade Bevel Angle  = arcsin( cos ( 45 )  *  cos( 36.8699 )) = 34.4499007767502°

Crown Molding Miter Angle  = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Crown Molding Miter Angle  = arctan( sin ( 45 ) ÷ tan( 36.8699 ))= 43.3138542102836°

Crown Molding Saw Blade Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Crown Molding Saw Blade Bevel Angle  = arcsin( cos ( 45 )  *  cos( 36.8699 )) = 34.4499007767502°

Rake Crown Molding Vertical Plane Angle = 90°
Rake Crown Spring Angle = 45°
Rake Crown Miter Angle = 35.2643896827547°
Rake Crown Bevel Angle = 30°

Hip Rafter Diamond Post Compound Angle = 25.6410063600364°
Hip Rafter Diamond Post Dihedral Angle = 34.1695442947226°

Hawkindale
S = 45.0000°
D = 36.8699°
R1 = 30.9638°
R2 = 28.0725°
R3 = 37.8636°
R4 = 48.8257°
R5 = 25.6410°
R6 = 78.8757°
R7 = 39.2962°
P1 = 43.3139°
P2 = 43.3139°
P3 = 26.5254°
P4 = 26.5254°
P5 = 27.9384°
P6 = 25.2011°
C1 = 34.4499°
C2 = 22.8337°
C3 = 34.4499°
C4 = 43.3139°
C5 = 34.4499°
A5 = 24.3052°
A7 = 38.6598°
A8 = 16.7885°
A9 = 18.7478°

```

```Tetrahedron Angles for 9/12 irregular hip roof

D Angle = 53.1301
A Angle = 36.8699
C Angle = 30.9638
E Angle = 30.9638
B Angle = 21.1002

90-D Angle = 36.8699
90-A Angle = 53.1301
90-C Angle = 59.0362
90-E Angle = 59.0362
90-B Angle = 68.8998

The dihedral angle is the angle measured between two planes.
Dihedral Angle = 137.7996

Polygonal Pyramid Sides = 4.88
Pyramid Slant Angle = 36.8699
Pitch Angle = 36.8699
Plan Angle  = 53.1301
Working Angle = 36.8852459016393

Hip Rafter Pitch Angle = arctan( tan( Pitch Angle ) * sin( Plan Angle ))
Hip Rafter Pitch Angle = arctan( tan( 36.8699 ) * sin( 53.1301 )) = 30.9637579168711°

Hip Rafter Backing Angle = arctan( sin( Hip Rafter Pitch Angle) ÷ tan( Plan Angle ) )
Hip Rafter Backing Angle = arctan( sin( 30.9637579168711) ÷ tan( 53.1301 ) ) = 21.1001984464996°

Hip Rafter Plumb Backing Angle = arctan( tan( Pitch Angle ) * cos( Plan Angle ))
Hip Rafter Plumb Backing Angle = arctan( tan( 36.8699 ) * cos( 53.1301 )) = 24.2277483279458°

Hip Rafter Side Cut Angle  = arctan( cos( Hip Rafter Pitch Angle ) ÷ tan( Plan Angle ))
Hip Rafter Side Cut Angle = arctan( cos( 30.9637579168711 ) ÷ tan( 53.1301 )) = 32.7458710613747°

Jack Rafter Side Cut = arctan( cos( Pitch Angle ) ÷ tan( Plan Angle ))
Jack Rafter Side Cut = arctan( cos( 36.8699 ) ÷ tan( 53.1301 )) = 30.9637579168711°

Roof Sheathing Angle = arccos( cos( Plan Angle ) * cos( Hip Rafter Pitch Angle ))
Roof Sheathing Angle = arccos( cos( 53.1301 ) * cos( 30.9637579168711 )) = 59.0362420831289°

Frieze Block Saw Miter Angle = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Frieze Block Saw Miter Angle = arctan( sin ( 36.8699 ) ÷ tan( 53.1301 )) = 24.2277483279458°

Frieze Block Saw Blade Bevel Angle = arctan( sin( Frieze Block Saw Miter Angle ) ÷ tan( Pitch Angle ))
Frieze Block Saw Blade Bevel Angle = arctan( sin( 24.2277483279458 ) ÷ tan( 36.8699 )) = 28.6854027655007°

Square Tail Fascia Miter Angle  = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Square Tail Fascia Miter Angle  = arctan( sin ( 36.8699 ) ÷ tan( 53.1301 )) = 24.2277483279458°

Square Tail Fascia Saw Blade Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Square Tail Fascia Saw Blade Bevel Angle  = arcsin( cos ( 36.8699 )  *  cos( 53.1301 )) = 28.6854027655007°

Hip Rafter Square Tail Miter Angle  = arctan (( cos( Pitch Angle) * sin( Pitch Angle ) * cos(Plan Angle)) ÷ tan( Plan Angle))
Hip Rafter Square Tail Miter Angle  = arctan (( cos( 36.8699) * sin( 36.8699 ) * cos( 53.1301 )) ÷ tan(53.1301)) = 12.188635145255°

Hip Rafter Square Tail Saw Blade Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Hip Rafter Square Tail Saw Blade Bevel Angle  = arcsin( cos ( 36.8699 )  *  cos( 53.1301 )) = 28.6854027655007°

90° - Hip Rafter Square Tail Side Cut Angle (Foot)
R3 equals P2 rotated by C5.
R3 equals Jack Rafter Side Cut Angle rotated by the Hip Rafter Backing Angle
R3 = arctan( cos ( Hip Rafter Backing Angle ) * tan( Jack Rafter Side Cut Angle ))
R3 = arctan( cos ( 21.1001984464996 ) * tan( 30.9637579168711 )) = 29.2388545837033°

Dormer Square Tail Fascia Miter Angle  = arctan( cos ( Dormer Pitch Angle ) ÷ tan( Main Roof Pitch Angle ))
Dormer Square Tail Fascia Miter Angle  = arctan( cos ( 36.8699 ) ÷ tan( 36.8699 )) = 46.8476069378739°
Dormer Square Tail Fascia Miter Angle on Fascia = 90° - 46.8476069378739° = 43.1523930621261°

Dormer Square Tail Saw Blade Bevel Angle  = arcsin( sin ( Dormer Pitch Angle ) *  cos( Main Roof Pitch Angle ))
Dormer Square Tail Saw Blade Bevel Angle  = arcsin( sin ( 36.8699 )  *  cos( 36.8699 )) = 28.6854027655007°

Rake Stud Saw Miter Angle = Hip Rafter Plumb Backing Angle = 24.2277483279458°
Rake Stud Saw Blade Bevel Angle = arctan(cos(Hip Rafter Plumb Backing Angle)* tan(Hip Rafter Pitch Angle))
Rake Stud Saw Blade Bevel Angle = atan(cos(24.2277483279458)* tan(30.9637579168711)) = 28.6854027655007°

Rake Plate Miter Angle = Jack Rafter Side Cut Angle = 30.9637579168711°
Rake Plate Saw Blade Bevel Angle = Roof Pitch Angle = 36.8699°

Purlin Miter Angle  = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Purlin Miter Angle  = arctan( sin ( 36.8699 ) ÷ tan( 53.1301 ))= 24.2277483279458°

Purlin Saw Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Purlin Saw Bevel Angle  = arcsin( cos ( 36.8699 )  *  cos( 53.1301 )) = 28.6854027655007°

Crown Molding Miter Angle  = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Crown Molding Miter Angle  = arctan( sin ( 36.8699 ) ÷ tan( 53.1301 ))= 24.2277483279458°

Crown Molding Saw Blade Bevel Angle  = arcsin( cos ( Pitch Angle ) *  cos( Plan Angle ))
Crown Molding Saw Blade Bevel Angle  = arcsin( cos ( 36.8699 )  *  cos( 53.1301 )) = 28.6854027655007°

Rake Crown Molding Vertical Plane Angle = 106.2602°
Rake Crown Spring Angle = 36.8699°
Rake Crown Miter Angle = 30.9637579168711°
Rake Crown Bevel Angle = 21.1001984464996°

Hip Rafter Diamond Post Compound Angle = 19.7988783552241°
Hip Rafter Diamond Post Dihedral Angle = 26.9814151018306°

Hawkindale
S = 36.8699°
D = 53.1301°
R1 = 30.9638°
R2 = 12.1886°
R3 = 29.2389°
R4 = 32.7459°
R5 = 19.7989°
R6 = -70.4746°
R7 = 20.1197°
P1 = 24.2277°
P2 = 30.9638°
P3 = 18.3085°
P4 = 26.5174°
P5 = 46.8476°
P6 = 11.2289°
C1 = 28.6854°
C2 = 10.6739°
C3 = 21.1002°
C4 = 30.9638°
C5 = 21.1002°
A5 = 17.9808°
A7 = 24.2277°
A8 = 4.4464°
A9 = 12.1886°
```

## Saturday, June 23, 2012

I made some router templates for the geometric symbols Ad Quadratum - Seed of Life - Ad Triangulum. These router templates were used for the window trim and soffit  knee braces.

The router templates weren't perfect enough so I had some custom router templates made out of 1/2" HDPE by www.routertemplate.net. Well it turns out that www.routertemplate.net is owned by Brenton Green who also owns www.digiconsoo.com. Digital Construction Solutions does CNC fabrication of residential and light commercial construction products.   Groin vaults, eyebrow and barrel roofs and a lot of other really neat geometrical construction related products. Check out his websites.

This picture is the base template, 18" square with a 12" square cut out.

It's pretty easy to align with the centering marks.

After the base is aligned with the centering marks the circle template is inserted into the base. The circle template is used for all three symbols.  The templates are setup for a 60° V-Groove router bit with a 5/8" guide bushing. This template has an 3 5/8" diameter hole to allow for the 3" circle.

Here's I'm routing the seed of life(daisy wheel). The template has 1/4" holes spaced around the equilateral triangle for a 1/4" dowel. The router base also has a 1/4" hole drilled into the router base to swing an 1 1/2" radius for the daisy wheel arcs.

Route the three arcs of the daisy wheel and then reverse the template to route in the other 3 arcs of the daisy wheel.

Here's the 4x4 redwood window header with 3 daisy wheels routed.

Here I was testing the custom router templates for the geometric symbols Ad Quadratum - Seed of Life - Ad Triangulum.

Picture of the knee brace and window trim routed.

Knee brace and window trim installed, except for the window header.

Some pictures of the finished product.

I tried using a diamond tip router bit to route the granite, but as you can see the diamond tip router bit didn't really work. After routing the Ad Triangulum into the granite the diamond tip router bit was diamond-less even through I used diamond coolant with the router bit.

Looks like the granite is too tough to route with the diamond tip router bit I had. (\$65.00)

## Tuesday, June 19, 2012

### Hexagon and Octagon Roof Framing Geometry

Tait Chappell told me about the advance timber framing class at Fox Maple they put on last week where the students built a hexagon 13.5/12 common pitch with two dormers at 22/12 valleys joining into rotated purlin headers. So I decided to draw out a similar hexagon plan. I couldn't let Tait have all the fun.

The day I started to draw out  the hexagon roof plan my son Brian told me a funny, but sad story that needs addressing. He's been working for a company this last month that builds commercial buildings, about 25,000 sqft a year for the last 10 years. So they know what their doing, right??? We'll their current job, a prevailing wage job at \$62.00 an hour, was bid too low so the owner of the company decided that he and his partner would cut and stack the octagon roof. Well, the owner cut the octagon hips as if it were a normal hip rafter and of course the hip rafters didn't fit. Then my son Brian explained to them that the octagon hip rafter rafters are not at the common rafter pitch and their not at the 90° eave angle hip rafter slope. He told them how to calculate the octagon hip rafters the old fashion way. Use the rise and the hip rafter run to calculate the octagon hip rafter slope angle.

So both of them had Construction Master Calculators. So why is it that they couldn't figure out the octagon hip rafter slope angle? Their ignorance? Or the Construction Master Calculator? It's 2012, why in the world world anyone call their calculator a "Master" when it can't even calculate an octagon hip rafter that the Romans and Greeks did 2,000+ years ago. Or the French carpenters did  500+ years ago.

Here's a drawing showing how to calculate the hip rafter slope angle for an octagon roof the old fashion way using a CMC calculator. Or they could have purchased a Chappell Master Framing Square that has the Hexagon and Octagon roof framing information printed on the square.

Using the Chappell Master Framing Square, line 1 under 8:12, .6159 . Framing Square usage: 6.159 on the tongue and 10 on the body to draw out the plumb and level cuts of the octagon hip rafter.
Or arctan(.6159) = 31.62892°

Using the Chappell Master Framing Square, line 2 under 8:12, 2.902 * 56" = 162.4854", it doesn't get any easier than that.

Using the Chappell Master Framing Square, line 3 under 8:12, 3.069 * 56" = 171.8648", it doesn't get any easier than that.

Here's the drawing I started working on. Hexagon roof with13.5: common rafter pitch and 22:12 dormers into a rotated purlin header. The hexagon was drawn out using Ad Triangulum ( equilateral triangle rotated within the circle). Just like the Romans and Greeks did 2,000+ years ago. Do the French carpenters of today use Ad Triangulum to draw out hexagons?

Rafter Tools is an android app with functions to calculate complex roof framing angles including Hexagon and Octagon roofs.

## Saturday, June 2, 2012

### Eyebrow & Barrel Roof Dormer Structural Design

I received a set of plans to bid on for the roof framing only, that has 18 barrel roofs on the 7600 SQ FT house. The architect did a great job of designing the house, however the structural design for the barrel roofs with rolled eaves was missing details on how to support the rolled eaves.I think the main reason we don't see a lot of eyebrow dormers in California is due to the lack of structural engineering for eyebrow dormers. I'm hoping the following drawings will correct that problem.

Typical structural design on the plans for the barrel roof with rolled eaves.

This is the shape of the barrel roof with rolled eaves.

This is the structural design I came up with to support the barrel vault roofs with rolled eaves. The internal ceiling will be a barrel vault with an elliptical valley and the exterior roof will be an eyebrow dormer with zero pitch.

For the bigger barrel roofs on this house this is the structural design I came up with to support the rolled eaves.

I've left some of the details out of these drawings for clarity. (Extra beams for the elliptical valley support)

One of the things on this set of plans that really impressed me was the 135° eave angle with a 8:12 pitch on one side of the valley and an 10:12 pitch on the other side of the valley. If this was an 90° eave angle it wouldn't be a big deal. I can figure out  the plan angles for this 135° eave angle, but I'm wondering how the architect knew how to draw it out correctly?

If you don't know the math for an eyebrow roof dormer, then contact Brent Green at DigCon and he'll make it for you.